A brief summary of the bounds on the solution of the algebraic matrix equations in control theory

Mori, Toshisada; Deresei, I. A.

doi:10.1080/00207178408933163

Cited by 103 publications

(43 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In practice, analytical solution of this equation is complicated, particularly when the dimensions of the system matrices are high. As such, a number of works have been presented over the past three decades for deriving solution bounds of this equation [2][3][4][5][7][8][9][10][11][12][13][14][15][16][17][18][19][20]22,[24][25][26], to reduce the computational burdens required to solve it analytically. Not only do solution bounds provide estimates for the solution of this equation, but they can also be applied to deal with practical situations involving the solution of this equation.…”

Section: Introductionmentioning

confidence: 99%

New upper solution bounds of the discrete algebraic Riccati matrix equation

Davies

Shi

Wiltshire

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

In this note, we present upper matrix bounds for the solution of the discrete algebraic Riccati equation (DARE). Using the matrix bound of Theorem 2.2, we then give several eigenvalue upper bounds for the solution of the DARE and make comparisons with existing results. The advantage of our results over existing upper bounds is that the new upper bounds of Theorem 2.2 and Corollary 2.1 are always calculated if the stabilizing solution of the DARE exists, whilst all existing upper matrix bounds might not be calculated because they have been derived under stronger conditions. Finally, we give numerical examples to demonstrate the effectiveness of the derived results.

show abstract

Section: Introductionmentioning

confidence: 99%

New upper solution bounds of the discrete algebraic Riccati matrix equation

Davies

Shi

Wiltshire

2008

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…where % A A; % P P; % Q Q 2 R nÂn ; if % A A is a stable matrix, we have the norm bound for % P P [22].…”

Section: Theorem 21mentioning

confidence: 98%

A new method for suboptimal control of a class of non-linear systems

Xin

Balakrishnan

2005

Optim. Control Appl. Meth.

View full text Add to dashboard Cite

SUMMARYIn this paper, a new non-linear control synthesis technique (y-D approximation) is discussed. This approach achieves suboptimal solutions to a class of non-linear optimal control problems characterized by a quadratic cost function and a plant model that is affine in control. An approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation is sought by adding perturbations to the cost function. By manipulating the perturbation terms both semi-global asymptotic stability and suboptimality properties are obtained. The new technique overcomes the large-control-for-large-initial-states problem that occurs in some other Taylor series expansion based methods. Also this method does not require excessive online computations like the recently popular state dependent Riccati equation (SDRE) technique. Furthermore, it provides a closed-form non-linear feedback controller if finite number of terms are taken in the series expansion. A scalar problem and a 2-D benchmark problem are investigated to demonstrate the effectiveness of this new technique. Both stability and convergence proofs are given.

show abstract

“…Davies et al [3] have also pointed out that instead of solving the CALE (1) for the solution matrix P, one can use solution bounds in place of the exact solution P to solve the optimization problem for a linear system. Besides, it is found that they can be applied to treat many control problems such as robust stability analysis for time-delay systems [17,25], robust root clustering [16,26], determination of the size of the estimation error for multiplicative systems [10], and so on [19]. In the literature, Gajic and Qureshi [4] also explained a motivation for studying the solution bounds but not the exact solution of the CALE (1).…”

Section: Consider the Continuous Algebraic Lyapunov Equation (Cale)mentioning

confidence: 99%

“…(28) leads to the bound (20) and substituting Eqs. (11) and (19) into Eq. (28) results in the bound (21).& Remark 1.…”

Section: Theorem 3 Ifmentioning

confidence: 99%

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

Lee

Chen

2009

Journal of the Franklin Institute

View full text Add to dashboard Cite

A brief summary of the bounds on the solution of the algebraic matrix equations in control theory

Cited by 103 publications

References 14 publications

New upper solution bounds of the discrete algebraic Riccati matrix equation

New upper solution bounds of the discrete algebraic Riccati matrix equation

A new method for suboptimal control of a class of non-linear systems

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

Contact Info

Product

Resources

About